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hasWLP -- Checks whether a graded Artinian algebra has the Weak Lefschetz Property (WLP)

Description

hasWLP(p,L) check if the monomial complete intersections $A=k[x_1,...,x_n]/(x_1^{a_1}, …, x_n^{a_n})$, where k is a field of characteristic p has the Weak Lefschetz property. For p=2 this method uses Theorem 8.1 in [KMRR,24]. For p>2, it uses the Han-Monsky multiplication obtained with Conjecture 4.1 [KMRR,25].

i1 : p = 3;
i2 : L = {3,3,3};
i3 : hasWLP(p, L)

o3 = false
i4 : p = 3;
i5 : L = {4,5,7};
i6 : hasWLP(p, L)

o6 = true

It is possible to check the WLP without employing the Han-Monsky multiplication, setting the option UseConjecture to be false.

i7 : p = 5;
i8 : L = {3,7,8}

o8 = {3, 7, 8}

o8 : List
i9 : hasWLP(p, L, UseConjecture =>false)

o9 = true

hasWLP(R,I) check if the graded Artinian algebra R/I, where R is a standard graded polynomial ring, has the Weak Lefschetz property.

i10 : R=QQ[x,y,z];
i11 : I=ideal(x^3, y^3, z^3, x*y*z);

o11 : Ideal of R
i12 : hasWLP(R,I)

o12 = false
i13 : R=QQ[x,y,z];
i14 : I=ideal(x*z, x*y, z^3,y^3,x^4+6*y^2*z^2);

o14 : Ideal of R
i15 : hasWLP(R,I)

o15 = true

The options GorensteinAlg and MonomialAlg allow simplifying the computation when the input I is respectively a Gorenstein or a monomial ideal.

i16 : R=QQ[x,y,z];
i17 : I=ideal(x^9,y^9,z^9,x^3*y^3*z^3);

o17 : Ideal of R
i18 : hasWLP(R,I, MonomialAlg => true)

o18 = false
i19 : R=ZZ/7[x,y,z];
i20 :  I=ideal(x*z, x*y, z^3,y^3,x^4+6*y^2*z^2);--Gorenstein

o20 : Ideal of R
i21 : hasWLP(R,I,GorensteinAlg => true)

o21 = true
i22 : R=ZZ/17[x,y,z];
i23 : I=ideal(3*x^4+y^4,z*y^3, x^5-z^5);--Gorenstein

o23 : Ideal of R

Caveat

The instances hasWLP(R,I) and hasWLP(R,I, GorensteinAlg=>true) require R to be a standard graded polynomial ring over a sufficiently large field k, e.g., R=QQ[x_1,…,x_n]. If R is a polynomial ring over a finite field, then when hasWLP(R, I) outputs false, it confirms that R/I fails the WLP. However, when hasWLP(R, I) outputs true, we can only conclude that R/I has the WLP over a field extension. The same holds for hasWLP(R,I, GorensteinAlg=>true).

This does not apply to hasWLP(p,L) and hasWLP(R,I, MonomialAlg=>true), which work over any field k.

Ways to use hasWLP:

  • hasWLP(PolynomialRing,Ideal)
  • hasWLP(ZZ,List)

For the programmer

The object hasWLP is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1467:0.